Elliptic surfaces over P 1 and large class groups of number fields
aa r X i v : . [ m a t h . N T ] M a y Elliptic surfaces over P and large class groups of number fields Jean Gillibert Aaron Levin ∗ May 2019
Abstract
Given a non-isotrivial elliptic curve over Q ( t ) with large Mordell-Weil rank, we explain howone can build, for suitable small primes p , infinitely many fields of degree p − p -torsion subgroup. As an example, we show the existence of infinitelymany cubic fields whose ideal class group contains a subgroup isomorphic to ( Z / Z ) . Our initial motivation for the present paper is the following conjecture on class groups ofnumber fields, which belongs to folklore, and is a consequence of the Cohen-Lenstra heuristics. If K is a number field, we denote by O K its ring of integers, and by Cl( K ) its ideal class group. Conjecture 1.1.
Let k be a number field, let p be a prime number, and let d > be an integer.Then dim F p Cl( L )[ p ] is unbounded when L/k runs through all extensions of degree [ L : k ] = d . When d = p , and more generally when p divides d , this conjecture follows from class field theory[Bru65, RZ69] (or see [Lev07, Th. 3.4] for a proof aligned with the techniques of the present paper).On the other hand, when p and d are coprime, there is not a single case where Conjecture 1.1 isknown to hold. For example, given a prime p , it is known [Yam70] that there exist infinitely manyimaginary quadratic fields L/ Q such that dim F p Cl( L )[ p ] ≥
2. For p ≥ p -ranks of class groups of quadratic fields.In the present paper, we investigate this conjecture in the specific case when d = p − k ( t ) withlarge rank, for almost all primes p one is able to produce a curve C which admits a morphism C → P k of degree p −
1, and whose Picard group has large p -rank (see Theorem 2.1). Thisconstruction allows one to use large Mordell-Weil rank over Q ( t ) to produce number fields ofdegree p − p -rank. In particular, we show the existence ofinfinitely many cubic fields whose ideal class group has 2-rank r ≥
11. This improves on a resultof Kulkarni [Kul18], who proved this statement with r ≥ Let k be a number field, and let E be an elliptic curve over k ( t ). We denote by E → P k theN´eron model of E over P k , by which we mean the group scheme model, which is the smooth locusof N´eron’s minimal regular model (see [BLR90, § § ∗ Supported in part by NSF grant DMS-1352407.
1y abuse of notation, we identify closed points of P k and discrete valuations of k ( t ). If v ∈ P k is such a point, we denote by E v the special fiber of E at v , and by Φ v the group of connectedcomponents of E v . The Tamagawa number of E at v is by definition the order of Φ v ( k v ), and wedenote it by c v . If E has good reduction at v , then Φ v = { } and c v = 1. The set of places of badreduction being finite, we have that c v = 1 for all but finitely many v . Theorem 1.2.
Let k be a number field and let E be an elliptic curve over k ( t ) . Assume that E does not have universal bad reduction at any prime of k (Definition 2.5). Let p be a prime numbersuch that E [ p ] is irreducible as a Galois module over k ( t ) .1) If p ≥ , there exist infinitely many field extensions L/k of degree p − such that dim F p Cl( L )[ p ] − dim F p Cl( k )[ p ] ≥ rk Z E ( k ( t )) − { v ∈ P k , p | c v } − rk Z O × L + rk Z O × k . (1)
2) If p = 2 , there exist infinitely many field extensions L/k of degree such that dim F Cl( L )[2] − dim F Cl( k )[2] ≥ rk Z E ( k ( t )) − { v ∈ P k , | c v } − rk Z O × L + rk Z O × k − { v ∈ P k , the red. type of E at v is I ∗ n for some n ≥ } . (2)The proof is given in § E does not have universal bad reductionat any prime of k is a technical condition detailed in § E , this condition can be verified numerically.If E is not isotrivial, then E [ p ] is irreducible as a Galois module over k ( t ) for all but finitelymany primes p . If in addition E does not have universal bad reduction at any prime of k , thenthe conclusion of Theorem 1.2 holds for all but finitely many primes p . Remark 1.3.
When p gets large, the term rk Z O × L also gets large: according to Dirichlet’s unittheorem, it is bounded below by p − −
1. Therefore, given an elliptic surface E , Theorem 1.2should be applied to primes p which are small with respect to the rank of E . Remark 1.4.
When p = 2, the condition that E does not have universal bad reduction at anyprime of k can be replaced by the condition that its N´eron model E has a fiber of type II, II ∗ , IVor IV ∗ at some v ∈ P ( k ). In fact, we obtain in this case a stronger version of (2), in which thecontribution from the units is removed; see Proposition 2.4. Remark 1.5.
The fields
L/k of degree p − L ′ with[ L ′ : k ] = p + 1. In specific situations, the equation of E being given, one may perform additionalcomputations in order to obtain a lower bound on dim F p Cl( L ′ )[ p ]. See Remark 2.2 for details. Applying Theorem 1.2 to an elliptic curve over Q ( t ) with large rank constructed by Kihara[Kih01], we obtain the following result (see § Theorem 1.6.
There exists a trigonal curve C , defined over Q , such that dim F Pic( C )[2] ≥ . Moreover, there exists a degree three morphism φ : C → P such that, for all but O ( √ N ) integers t ∈ Z with | t | ≤ N , the field of definition of φ − ( t ) is a cubic field K t with exactly one real place,satisfying dim F Cl( K t )[2] ≥ . emark 1.7. Nakano [Nak88] proved that there are infinitely many cubic fields K whose idealclass group has 2-rank at least 6. Recently, Kulkarni [Kul18] has improved on this result, obtaininginfinitely many cubic fields K whose ideal class group has 2-rank at least 8. To our knowledge,this is the best previously known result on the existence of an infinite family of cubic fields whoseclass group has large 2-rank. In fact, our method is closely related to Kulkarni’s, in the sense thatin both cases one considers trigonal curves which come from the 2-torsion of an elliptic fibration. Picard groups of curves over finite fields are considered natural analogues of class groups ofnumber fields. Thus, we include in this section a remark on the construction of curves over finitefields whose Jacobian contains a large torsion subgroup.In contradistinction to Conjecture 1.1, this geometric analogue does not appear to have beenthe subject of intense investigation. Nevertheless, as in the number field case, one can make thefollowing remark: the rational p -torsion subgroup of Jacobians of curves of gonality p can be madeas large as possible by considering equations of the form y p = f ( x ), where f splits as a productof linear factors. It seems harder to find curves of gonality coprime to p whose Jacobian containsa rational p -torsion subgroup which is large (relative to the gonality). We shall give some resultstowards this goal.Ulmer has shown [Ulm02] that, given a prime q , one can find non-isotrivial elliptic curves over F q ( t ) with arbitrarily large Mordell-Weil rank. More precisely, given an integer n ≥
1, he considersthe elliptic curve E n defined by the equation y + xy = x + t d , where d = q n + 1 , ( E n )and he proves that the rank of E n over F q ( t ) is at least ( q n − / n .Let us assume that q ≥
5. Then the trigonal curve C ,n → P describing the 2-torsion of E n isdefined by the equation 4 x + x − t d = 0 . ( C ,n )We claim that C ,n is geometrically irreducible. Indeed, if the equation above had a root x in F q ( t ), then by Gauss’s lemma this root would be a polynomial which divides t d , and this leads toa contradiction.Applying Theorem 2.1 to the elliptic curves E n , we obtain in § Theorem 1.8.
Let q ≥ be a prime. Then1) The family of trigonal curves C ,n defined above (over F q ) satisfies dim F Pic( C ,n )[2] ≥ q n − n − .
2) Given an integer n ≥ , for all but finitely many primes p , there exists a curve C over F q which admits a morphism C → P of degree p − , such that dim F p Pic( C )[ p ] ≥ q n − n − . Remark 1.9.
Using similar techniques, it has been proved in [ ˇCes15, Section 5] that, if p and q are two distinct primes, there exists a constant m depending only on p such that the size ofPic( C )[ p ] is unbounded when C runs through hyperelliptic curves over F q m .3 Proofs P k In this section, we briefly recall the main result of [GL18] in the setting of elliptic surfaces over P k . We refer the reader to loc. cit. for further details and comments.Let k be a perfect field of characteristic not 2 or 3. In the applications we have in mind, k maybe a number field, or a finite field, or the algebraic closure of such fields.Let E be an elliptic curve over k ( t ), and let E → P k be its N´eron model. If v is a closed pointof P k , we let E v , Φ v , and the Tamagawa number c v be as in § P k whose fiber at v is Φ v , and by E p Φ the inverse image of p Φ by the natural map
E →
Φ.Given a prime number p = char( k ), we denote by E [ p ] → P k the group scheme of p -torsionpoints of E . Because p = char( k ) the multiplication-by- p map on E , which is a smooth groupscheme over P k , is ´etale, and in particular E [ p ] → P k is ´etale. On the other hand, E [ p ] → P k is quasi-finite, but not finite in general. This is why we consider the smooth compactification of E [ p ] \ { } , that we denote by C , endowed with its canonical finite map C → P k of degree p − E [ p ] \ { } coincides with the ´etale locus of C → P k ; we refer to [GL18,Proof of Lemma 2.8] for further details.The following result is a special case of [GL18, Theorem 1.1]. Assuming that C is geometricallyintegral, it provides an upper bound on the rank of E in terms of the Tamagawa numbers c v andthe p -torsion in the Picard group of C . Its proof relies on p -descent techniques, analogous to thenumber field case. Theorem 2.1.
Let p = char( k ) be a prime number, and let C be the smooth compactification of E [ p ] \ { } . Assume that C is geometrically integral, or equivalently that E [ p ] is irreducible as aGalois module over k ( t ) . Then1) There is an injective morphism E p Φ ( P k ) /p E ( P k ) −→ Pic( C )[ p ] . (3)
2) If p ≥ , we have dim F p Pic( C )[ p ] ≥ dim F p E p Φ ( P k ) /p E ( P k ) ≥ rk Z E ( k ( t )) − { v ∈ P k , p | c v } , (4) where c v denotes the Tamagawa number of E at v .3) If p = 2 , then dim F Pic( C )[2] ≥ dim F E ( P k ) / E ( P k ) ≥ rk Z E ( k ( t )) − { v ∈ P k , | c v }− { v ∈ P k , the red. type of E at v is I ∗ n for some n ≥ } . (5)In fact, the injective morphism (3) is obtained by composing maps E p Φ ( P k ) /p E ( P k ) −−−−→ H ( C, µ p ) −−−−→ Pic( C )[ p ] (6)4he first being obtained by Kummer theory on E , and the second being the natural one. Thecohomology group in the middle is computed with respect to the ´etale topology, or equivalentlythe fppf topology since µ p is ´etale over C under the assumptions of Theorem 2.1.A comment on the terminology: when we say that E has a fiber of type I ∗ n at v , we meanit over k v , and not just over k . More precisely, this means that the Kodaira type of E v over k isI ∗ n , and that the four components of E v are rational over k v , in other terms Φ v ( k v ) ≃ ( Z / . Ingeneral, the reduction type at v can be described by the data of the reduction type over k togetherwith the action of the absolute Galois group of k v on Φ v . See Liu’s book [Liu02, § Remark 2.2.
The curve C in the statement of Theorem 2.1 corresponds to the field over which E acquires one rational p -torsion point. One can also introduce a curve C ′ corresponding to thefield over which E has a rational cyclic subgroup of order p ; then we have canonical maps C −→ C ′ −→ P k of degree p − p +1, respectively. Given a specific example of a curve E , one can find equationsfor C and C ′ , and compute the kernel of the norm map Pic( C )[ p ] → Pic( C ′ )[ p ]. If this kernel issmall enough then, using techniques of § C ′ → P k of degree p + 1 inorder to build extensions L ′ /k of degree p + 1 with dim F p Cl( L ′ )[ p ] large. Proof of Theorem 1.8.
Ulmer has checked that E n has reduction I at places dividing (1 − t d ),and has split multiplicative reduction I d at t = 0. The last place of bad reduction is t = ∞ , wherethe possible reduction types are I , II, II ∗ , IV, IV ∗ or I ∗ depending on q n + 1 (mod 6). Then 1)is a consequence of (5) in Theorem 2.1, the error term − − −
2, where − t = 0 whose Tamagawa number is divisible by two, and − ∗ . The second statement follows similarlyfrom (4) in Theorem 2.1, combined with the following observation: the elliptic curve E n is notisotrivial, hence according to the geometric version of Shafarevich’s theorem, E n [ p ] is k -irreduciblefor all but finitely many primes p . II , II ∗ , IV , IV ∗ Let k be a number field, and let E be an elliptic curve over k ( t ) defined by a Weierstrassequation y = x + a ( t ) x + b ( t ) . Let us assume that E ( k ( t ))[2] = { } , which is equivalent to saying that E [2] is irreducible asa Galois module over k ( t ). Then the smooth compactification of E [2] \ { } is none other than thesmooth, projective, geometrically integral curve C defined by the affine equation x + a ( t ) x + b ( t ) = 0 , and the canonical map C → P k is just the t -coordinate map, which has degree 3. Lemma 2.3.
Assume
E → P k has a fiber of type II , II ∗ , IV or IV ∗ at some v ∈ P k . Then the t -coordinate map C → P k is totally ramified above v .Proof. This follows from the proof of Lemma 2.8 in [GL18].5 roposition 2.4.
Let E be an elliptic curve over k ( t ) such that E ( k ( t ))[2] = { } . Assume inaddition that its N´eron model has a fiber of type II , II ∗ , IV or IV ∗ at some v ∈ P ( k ) . Then thereexist infinitely many cubic field extensions L/k such that dim F Cl( L )[2] − dim F Cl( k )[2] ≥ rk Z E ( Q ( t )) − { v ∈ P k , | c v }− { v ∈ P k , the red. type of E at v is I ∗ n for some n ≥ } . (7) Proof.
Let C be the trigonal curve defining 2-torsion points as above. According to Lemma 2.3,the natural trigonal map C → P k is totally ramified over some rational point. According to [BG18,Theorem 1.4], there exist infinitely many number fields L/k of degree 3 such thatdim F Cl( L )[2] − dim F Cl( k )[2] ≥ dim F Pic( C )[2] . The result then follows from the last statement of Theorem 2.1.
We now examine proving inequalities as in Proposition 2.4 in the absence of fibers of type II,II ∗ , IV or IV ∗ , and working with any prime number p .For that purpose, we first introduce the notion of universal bad reduction for an elliptic familydefined over a number field.Let k be a number field, and let E be an elliptic curve over k ( t ) defined by a Weierstrassequation y = x + a ( t ) x + b ( t )with a ( t ) , b ( t ) ∈ O k [ t ]. Let E → P k be the N´eron model of E , and let Σ ⊂ P k be the set of placesof bad reduction of E . Definition 2.5.
Let p be a prime ideal in O k . We will say that p is a prime of universal badreduction for E if p is a prime of bad reduction for every elliptic fiber E t , t ∈ P ( k ) \ Σ. Remark 2.6.
Let ∆( t ) := − a ( t ) + 27 b ( t ) ) be the discriminant of the Weierstrass equationdefining E . If p is a prime of universal bad reduction for E , then ∆( t ) (mod p ) is divisible by t q − t , where q = N ( p ) is the (absolute) norm of p . In particular, if ∆( t ) (mod p ) is not identicallyzero, then we must have N ( p ) ≤ deg ∆.Before we state our main result, we set some notation: if R is a commutative ring, we denoteby H ( R, µ p ) the fppf cohomology group H (Spec( R ) , µ p ). If K is a number field or a local field,then we identify H ( O K , µ p ) with a subgroup of H ( K, µ p ) via the natural restriction map. Theorem 2.7.
Suppose that E does not have universal bad reduction at any prime of k . Assumethat E [ p ] is an irreducible Galois module over k ( t ) . Let C be the smooth compactification of E [ p ] \ { } , and let H denote the image of the injective map from (6) E p Φ ( P k ) /p E ( P k ) ֒ → H ( C, µ p ) . Then there exists a map ψ : C → P k of degree p − such that, for all but O ( √ N ) integers t ∈ Z with | t | ≤ N , we have that:1) P t,ψ := ψ − ( t ) is the spectrum of a field k ( P t,ψ ) , with [ k ( P t,ψ ) : k ] = p − ;2) the image of H under the specialization map P ∗ t,ψ : H ( C, µ p ) → H ( k ( P t,ψ ) , µ p ) lands intothe subgroup H ( O k ( P t,ψ ) , µ p ) ; ) the specialization map P ∗ t,ψ above is injective on H .Proof. Let X , . . . , X r be independent µ p -torsors over C generating H . Then by the Chevalley-Weiltheorem, there exists a finite set T of places of k such that the X i can be extended to µ p -torsorsin H ( C T , µ p ), where C T denotes a smooth projective model of C over Spec( O k,T ). In particular, T contains the set of bad places of C .We denote by φ : C → P k the natural map ( x, t ) t , of degree p −
1. For each t ∈ P ( k ), welet P t := ψ − ( t ). It follows from the construction of T and the projectivity of C T that, for each t ∈ P ( k ), the image of H under the specialization map P ∗ t : H ( C, µ p ) → H ( k ( P t ) , µ p ) lands intothe subgroup H ( O k ( P t ) ,T , µ p ).Let us now pick p ∈ T . By assumption, E does not have universal bad reduction at p , andhence there exists t p ∈ P ( k ) such that E t p has good reduction at p .Let N t p be the N´eron model of E t p over Spec( O k, p ). Then N t p is an abelian scheme and itfollows that the map [ p ] : N t p → N t p is an epimorphism for the fppf topology, regardless of theresidue characteristic of p .Let q | p be a place of k ( P t p ) above p , and let P q ∈ C ( k ( P t p ) q ) be the corresponding localizationof P t p . More geometrically, P t p ⊗ k p is the disjoint union of the P q , hence ( E t p [ p ] \ { } ) ⊗ k p is theunion of the P q . The Weil pairing induces a map w : N t p [ p ] −→ Y q | p Res O k ( Pt ) , q / O k, p µ p . We have a commutative diagram E t p ( k p ) /p N t p ( O k, p ) /p −−−−→ Q q | p H ( O k ( P t ) , q , µ p ) t ∗ p x y E p Φ ( P k ) /p E ( P k ) −−−−→ H ( C, µ p ) ( P t p ⊗ k p ) ∗ −−−−−−→ Q q | p H ( k ( P t ) q , µ p )in which the upper right map comes from fppf Kummer theory on the abelian scheme N t p , combinedwith the Weil pairing map. The vertical left map is obtained by specializing the elliptic family at t p ∈ P ( k p ).This proves that, under the map P ∗ t p , all elements of H (and, in fact, all µ p -torsors built fromKummer theory on E ) specialize to torsors which are integral at all places of k ( P t ) above p . By avariant of Krasner’s lemma, similar to [BG18, Lemma 2.3], the same holds for any t ∈ P ( k ) whichis p -adically close enough to t p .Finally, let us recall that an element of H ( O k ( P t ) ,T , µ p ) belongs to H ( O k ( P t ) , µ p ) if and onlyif it belongs to H ( O k ( P t ) , q , µ p ) for each prime q ∈ T k ( P t ) . Putting everything together, we mayconclude that, if t ∈ P ( k ) is p -adically close enough to t p for every p in T , then the image of H under P ∗ t lands into H ( O k ( P t ) , µ p ).Let T Z denote the set of prime numbers lying below primes in T , and let ψ : C → P k be themap defined by ψ := 1( Q p ∈ T Z p ) M ( φ − t ) , where t ∈ k is p -adically close enough to t p for every p ∈ T , and M is a sufficiently large positiveinteger. The conclusion of the theorem follows from a quantitative version of Hilbert’s irreducibilityTheorem (see [GL12, Theorem 2.1]) applied to the composite map X × C · · · × C X r −−−−→ C ψ −−−−→ P k . (8)7n this final step we implicitly use the fact that X × C · · · × C X r is geometrically irreducible,which can be proved as follows: H injects into Pic( C ) according to the first statement of Theo-rem 2.1, and this remains true over k by injectivity of the map Pic( C ) → Pic( C × k k ). Hence H injects into H ( C × k k, µ p ) via the natural base-change map; in other terms, X , . . . , X r remainindependent over k . Proof of Theorem 1.2.
Given the inequalities (4) and (5) from Theorem 2.1, the result followsfrom Theorem 2.7 combined with the following fact: when applying Hilbert’s irreducibility the-orem to the cover (8), it is always possible to ensure that the field extension obtained is lin-early disjoint from a given extension fixed in advance. If one fixes the latter extension to bethe compositum of the fields that correspond to torsors in H ( O k , µ p ), then the natural map H ( O k , µ p ) → H ( O k ( P t,ψ ) , µ p ) is injective, and it follows from Kummer theory (see the proof ofTheorem 1.3 in [BG18]) thatdim F p H ( O k ( P t,ψ ) , µ p ) /H ( O k , µ p ) = dim F p Cl( k ( P t,ψ ))[ p ] − dim F p Cl( k )[ p ]+ rk Z O × k ( P t,ψ ) − rk Z O × k . To conclude, it suffices to point out that the image of the subgroup H from Theorem 2.7 by themap P ∗ t,ψ injects into the quotient group on the left. Let E be an elliptic curve over Q ( t ) without a rational 2-torsion point over Q ( t ). We mayassume that E is defined by an equation y = x + a ( t ) x + b ( t ) (9)where a ( t ) and b ( t ) belong to Z [ t ].We denote by C the smooth projective curve with affine equation x + a ( t ) x + b ( t ) = 0, whichis the smooth compactification of E [2] \ { } . We also let ∆( t ) := − a ( t ) + 27 b ( t ) ) be thediscriminant of the Weierstrass equation (9).Using an appropriately chosen elliptic curve E of large Q ( t )-rank, we shall apply Theorem 1.2to obtain infinitely many cubic fields with a class group of large 2-rank.In [Kih01], Kihara gives an example of an elliptic curve E over Q ( t ) of rank at least 14. Acalculation shows that in Kihara’s example, E contains singular fibers (over P Q ) of types I , I (two fibers), and I . The elliptic curve E is obtained as a specialization of a 3-dimensional familyof elliptic curves having rank at least 12. By using a different specialization of this family, wemay obtain a more advantageous singular fiber configuration (with Theorem 1.2 in mind) at theexpense of (possibly) lowering the Q ( t )-rank.Specifically, let E ′ be the elliptic curve over Q ( p, q, u ) of rank at least 12 constructed in [Kih01](in fact, Kihara gives a genus one quartic with 13 Q ( p, q, u )-points; one must choose one of the 13points as a base point for the elliptic curve). Let E be the elliptic curve over Q ( t ) obtained from E ′ by the specialization p = t , q = t + 6, u = t + 1. Then one can check numerically that there isno prime of universal bad reduction for E .We may write the equation of E in the form (9), where the discriminant ∆( t ) is irreducible in Q [ t ] of degree 96. There is exactly one bad place in P Q , where E has a singular fiber of type I .By specializing Kihara’s points, at say t = 1, and computing the associated height pairing, itis easily verified that E has rank at least 12 over Q ( t ).8hen, according to Theorem 2.1, we havedim F Pic( C )[2] ≥ rk Z E ( Q ( t )) ≥ . One also checks that there exists a rational value of t for which the equation x + a ( t ) x + b ( t ) = 0has a single real root. By variants of Theorem 2.7 (resp. Theorem 1.2) taking into account theplaces at infinity, it follows that there exists a trigonal morphism φ : C → P such that for all but O ( √ N ) integers t ∈ Z with | t | ≤ N , Q ( P t ) is a cubic number field with exactly one real place anddim F Cl( Q ( P t )) ≥ rk Z E ( Q ( t )) − rk Z O × Q ( P t ) ≥ . Remark 2.8.
One may possibly go further using famous constructions of Elkies. More precisely,Elkies describes in [Elk07] constructions of elliptic curves over Q ( t ) of ranks 17 and 18. The detailsof these constructions (e.g., explicit equations for the curves) remain unpublished. This exampleis likely to lead to further applications of our techniques. Acknowledgments
The first author was supported in part by the CIMI Excellence program while visiting the
Centro di Ricerca Matematica Ennio De Giorgi during the autumn of 2017. The second authorwas supported in part by NSF grant DMS-1352407.
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